Theorem abrexco 5463

Description: Composition of two image maps C(y) and B(w). (Contributed by set.mm contributors, 27-May-2013.)

Hypotheses

Ref	Expression
abrexco.1	⊢ B ∈ V
abrexco.2	⊢ (y = B → C = D)

Assertion

Ref	Expression
abrexco	⊢ {x ∣ ∃y ∈ {z ∣ ∃w ∈ A z = B}x = C} = {x ∣ ∃w ∈ A x = D}

Distinct variable groups:   y,A,z   y,B,z   w,C   y,D   x,w,y   z,w

Allowed substitution hints:   A(x,w)   B(x,w)   C(x,y,z)   D(x,z,w)

Proof of Theorem abrexco

Step	Hyp	Ref	Expression
1		df-rex 2620	. . . 4 ⊢ (∃y ∈ {z ∣ ∃w ∈ A z = B}x = C ↔ ∃y(y ∈ {z ∣ ∃w ∈ A z = B} ∧ x = C))
2		vex 2862	. . . . . . . 8 ⊢ y ∈ V
3		eqeq1 2359	. . . . . . . . 9 ⊢ (z = y → (z = B ↔ y = B))
4	3	rexbidv 2635	. . . . . . . 8 ⊢ (z = y → (∃w ∈ A z = B ↔ ∃w ∈ A y = B))
5	2, 4	elab 2985	. . . . . . 7 ⊢ (y ∈ {z ∣ ∃w ∈ A z = B} ↔ ∃w ∈ A y = B)
6	5	anbi1i 676	. . . . . 6 ⊢ ((y ∈ {z ∣ ∃w ∈ A z = B} ∧ x = C) ↔ (∃w ∈ A y = B ∧ x = C))
7		r19.41v 2764	. . . . . 6 ⊢ (∃w ∈ A (y = B ∧ x = C) ↔ (∃w ∈ A y = B ∧ x = C))
8	6, 7	bitr4i 243	. . . . 5 ⊢ ((y ∈ {z ∣ ∃w ∈ A z = B} ∧ x = C) ↔ ∃w ∈ A (y = B ∧ x = C))
9	8	exbii 1582	. . . 4 ⊢ (∃y(y ∈ {z ∣ ∃w ∈ A z = B} ∧ x = C) ↔ ∃y∃w ∈ A (y = B ∧ x = C))
10	1, 9	bitri 240	. . 3 ⊢ (∃y ∈ {z ∣ ∃w ∈ A z = B}x = C ↔ ∃y∃w ∈ A (y = B ∧ x = C))
11		rexcom4 2878	. . 3 ⊢ (∃w ∈ A ∃y(y = B ∧ x = C) ↔ ∃y∃w ∈ A (y = B ∧ x = C))
12		abrexco.1	. . . . 5 ⊢ B ∈ V
13		abrexco.2	. . . . . 6 ⊢ (y = B → C = D)
14	13	eqeq2d 2364	. . . . 5 ⊢ (y = B → (x = C ↔ x = D))
15	12, 14	ceqsexv 2894	. . . 4 ⊢ (∃y(y = B ∧ x = C) ↔ x = D)
16	15	rexbii 2639	. . 3 ⊢ (∃w ∈ A ∃y(y = B ∧ x = C) ↔ ∃w ∈ A x = D)
17	10, 11, 16	3bitr2i 264	. 2 ⊢ (∃y ∈ {z ∣ ∃w ∈ A z = B}x = C ↔ ∃w ∈ A x = D)
18	17	abbii 2465	1 ⊢ {x ∣ ∃y ∈ {z ∣ ∃w ∈ A z = B}x = C} = {x ∣ ∃w ∈ A x = D}

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861

This theorem is referenced by: (None)